Geodesic
On rational directions in the flat lattice
Čebyševskij sbornik, Tome 16 (2015) no. 2, pp. 273-281.

Voir la notice de l'article provenant de la source Math-Net.Ru

Rational and irrational rotations for the set of rational directions in the flat point lattice are considered. It is proved that in the case of rational rotations an order of noncrystallographic turn can be only 8 or 12. The set of rational directions in the rectangular point lattice with metric quadratic form $x^2+\lambda^2y^2$ and arbitrary its centering has irrational rotation if and only if the number $\lambda^2$ is rational. Bibliography: 3 titles.
Keywords: lattice, unit cell, rational direction, rotation, angle, tangent. 
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M. I. Shtogrin. On rational directions in the flat lattice. Čebyševskij sbornik, Tome 16 (2015) no. 2, pp. 273-281. http://geodesic.mathdoc.fr/item/CHEB_2015_16_2_a16/

[1] Ryshkov S. S., Fundamentals of the theory of point lattices and systems Delone, Publishing house of the Moscow State University, M., 2014, 142 pp.

[2] Gadolin A. B., “Displays all crystallographic systems and their units of one common origin”, Notes of the Imperial St. Petersburg Mineralogical Society. The second series, 4, 1867, 112–200

[3] Shubnikov A. V., Koptsik V. A., Symmetry in Science and Art, ed. D. Harker, Plenum Press, New York–London, 1974, 420 pp.